We are particularly motivated in our research by applications of high-intensity ultrasonic waves, whose behavior is inherently nonlinear and modeled by nonlinear wave equations of this type. These applications range from non-invasive treatments of different medical disorders to non-destructive material testing and urge the need to accurately model and analyze ultrasonic waves in different settings. The thermoviscous dissipation of fluids is known to play an important role in the propagation of sound waves. In classical acoustic models, such as the Westervelt and Kuznetsov equations, these effects are encompassed within the medium parameter \(b\) , often referred to as the sound diffusivity; see [1]}, [2]}, [3]}, [4]}. Thus, the present work can also be understood as an investigation into how the behavior of solutions to nonlinear acoustic equations that model ultrasonic propagation changes with the sound diffusivity. We refer to Section  below for further details on modeling in nonlinear acoustics. Formally, setting \(b =0\)  in (REF ) and (REF ) corresponds to lossless nonlinear sound propagation through inviscid fluids or gases; see, e.g., [5]} and the references therein. Corresponding to the applications in nonlinear acoustics, we here focus on quadratic nonlinearities
\(f(p_t)=kp_t^2, \quad \alpha (p)=1-kp\) 
